The issue of what happens with a proposition in case of a failed presuppositions is still not settled. There is a huge pile of literature on this matter, so let me just quickly sketch the three main positions, using the classic case of:
(1) The present king of France is bald.
According to Russell, (1) is just false, if the presupposition that there is a present king of France is false. This is still a position that sometimes is maintained (especially for definite description as in (1)), but one must note that this does not really treat the inference that there is a present king of France as a presupposition. Instead it is treated as an entailment.
According to Strawson, a presupposition failure deprives a sentence of it ability to have a truth value. That is, if there is no King of France, (1) is neither true nor false. This often is modeled using a trivalent logic (e.g Beaver & Krahmer 2001), with the values 1 (true), 0 (false), and * (undefined).
According to the “dynamic” view (Stalnaker 1978), a failed preuspposition prevents a sentence from updating the so-called common ground (= the shared belief system of the interlocutors) and may lead to a failed state. That is, if I utter (1) in a conversation, the shared belief system should contain the proposition (or imply it; depending on the formal elaboration) that there is a king of France. If it does, the common ground is updated with the new information that the king of France is bald. If it does not contain the proposition that there is a king of France, (1) will either be unable to update the common ground or it will, but the result will be failed state.
Still the best overview over the historical development of the (technical) of presupposition is provided in the first part of Beaver 2001, which you can access here. A more concise overview (by Beaver & Geurts) can be found in the Stanford Encyclopedia of Philosophy. The Ohio State University also hosts an extensive bibliography on presuppositions.
Some important references
Beaver, David I. (2001): Presupposition and Assertion in Dynamic Semantics. Stanford: CSLI. http://semanticsarchive.net/Archive/jU1MDVmZ/book-2001.pdf.
Beaver, David I. & Emiel Krahmer (2001): “A partial account of presupposition pro- jection”. Journal of Logic, Language and Information 10.2, 147–182. doi: 10.1023/a: 1008371413822.
Beaver, David I. and Bart Geurts (2011): “Presupposition”. The Stanford Encyclopedia of Philosophy. Winter 2014. http://plato.stanford.edu/archives/win2014/entries/presupposition/
Lewis, David (1979): “Scorekeeping in a language game”. Journal of Philosophical Logic 8, 339–359.
Russell, Bertrand (1905): “On denoting”. Mind 14.56, 479–493. url: http://www.jstor. org.ubproxy.ub.uni- frankfurt.de/stable/2248381.
van der Sandt, Rob (1992): “Presupposition Projection as Anaphora Resolution”. Journal of Semanticsition projection as anaphora resolution 9.4, 333–377. doi: 10.1093/jos/9. 4.333.
Strawson, Peter F. (1950): “On referring”. Mind 59, 320–44.
Stalnaker, Robert (1978): “Assertion”. In: Peter Cole, ed.: Pragmatics. Syntax and Seman- tics 9. New York: Academic Press, 315–332.